Example 1:Find the general antiderivatives of each of the following using you knowledge of how to find derivatives.

a) 2f x x b) f x x c) 4

72

3F x x d) 2

1g x

x e) cos

dyx

dx

Example 2:

Find all functions g such that g x42

4sinx x

xx

x .

Example 3:

Solve the differential equation f x 3x2 if f 2 3 . Find both the general and particular solutions.

Example 4:

Find the particular solution to the following differential equation if 12x 20 1dy

e xdx

and y 0 2 .

Example 5:

Find the particular solution to the following differential equation if 2

22

12 6d y

xdx

x 4 and

a) y1 3 and y 0 6 b) y 0 4 and y 1 1.

Example 6:

a) Evaluate2

sin

cos

xdx

x b) Evaluate tan2 p 4dp

Show all work. No Calculator

Multiple Choice

1. If 2( ) 12 6 1f x x xʹ′ = − + , (1) 5f = , then (0)f equals(A) 2 (B) 3 (C) 4 (D) –1 (E) 0

2. Find all functions g such that ( )25 4 5x xg xx

+ +ʹ′ =

(A) ( ) 2 42 53

g x x x x C⎛ ⎞= + − +⎜ ⎟⎝ ⎠

(B) ( ) 2 42 53

g x x x x C⎛ ⎞= + + +⎜ ⎟⎝ ⎠

(C) ( ) ( )22 5 4 5g x x x x C= + − + (D) ( ) 2 4 53

g x x x x C⎛ ⎞= + + +⎜ ⎟⎝ ⎠

(E) ( ) ( )25 4 5g x x x x C= + + +

3. Determine ( )f t when ( ) ( )2 3 1f t tʹ′ʹ′ = + and ( )1 3f ʹ′ = , ( )1 5f = .

(A) ( ) 3 23 2 2 2f t t t t= − + + (B) ( ) 3 22 2 4f t t t t= − + +

(C) ( ) 3 23 2 3f t t t t= + − + (D) ( ) 3 2 2 3f t t t t= − + +

(E) ( ) 3 2 2 5f t t t t= + − +

4. Consider the following functions:

I. ( )2

1sin2xF x =

II. ( )2cos24xF x = −

III. ( )2

3cos2xF x = −

Which are antiderivatives of ( ) sin cos ?f x x x= (Hint: take the derivative of each and manipulate) (A) II only (B) I only (C) I & III only (D) I, II, & III (E) I & II only

5. A particle moves along the x-axis so that its acceleration at time t is ( ) 8 8a t t= − in units of feet andseconds. If the velocity of the particle at 0t = is 12 ft/sec, how many seconds will it take for theparticle to reach its furthest point to the right?

(A) 6 seconds (B) 5 seconds (C) 3 seconds (D) 7 seconds (E) 4 seconds

Free Response

6. Evaluate the following:

(a) 3 2 1x x dx⎛ ⎞+ +⎜ ⎟⎝ ⎠∫ (b)

3

42 3x x dxx

⎛ ⎞+ −⎜ ⎟⎜ ⎟⎝ ⎠∫ (c) ( )222 1t dt−∫

(d) ( )2 2sec csc cot dθ θ θ θ θ+ −∫ (e)2

cos1 cos

x dxx

⎛ ⎞⎜ ⎟−⎝ ⎠

∫ (f) ( )cos 3xx dx+∫

7. Solve the following differential equations. Find the general solution, then find the particular solutionusing the initial condition.

(a) ( ) 4f x xʹ′ = , ( )0 6f = (b) ( ) 38 5h t tʹ′ = + , ( )1 4h = − (c) ( ) 2f xʹ′ʹ′ = , ( )2 5f ʹ′ = , ( )2 10f =

(d) ( ) 3/2f x x−ʹ′ʹ′ = , ( )4 2f ʹ′ = , ( )0 0f = (e) ( ) sinf x xʹ′ʹ′ = , ( )0 1f ʹ′ = , ( )0 6f =

ANTIDERIVATIVES & INITIAL VALUE PROBLEMS PRACTICE

1)

A) 3p2x + c

B) 0

C) p3x + c

D) 3p2 + c

E) + c

2)

A) - + + c

B) 5x5 - 4x4 + 3x3 + c

C) - 3x2 + + c

D) 4x3 - 3x2 + 2x + c

E) - + + c

3)

A) - 2x2 + c

B) * + - x2 + c

C) -3x2 + 2x - 2 + c

D) * + - x2 + 2x + c

E) ( - 2x)(x - x2) + c

4)

A) ( )3 + cB) + c

C) (x2 - 2)3 + c

D) ( - 2x)2 + cE) - + 4x + c

5)

A) - + c

B) ( - 2x) + cC) + c

D) 3x2( - 2x) + cE) 6x5 - 6x2 + c

6)

A) + c

B) + c

C) 18x6 + 8x2 - 2x + c

D) 3x4 + 2x2 - x + c

E) x4 + x2 - x + c

7)

A) x  x + c

B) x  x + c

C) * x  x + c

D) 2  x + c

E)   x + c

8)

A)

B)

C)

D)

E)

9)

A)

B)

C)

D)

E)

10)

A) - + c

B)   x4 3 -   x3 + c

C) x   x4 3 - 6   x3 + c

D) x   x4 3 - x   x

3 2 + c

E) x   x4 3 - x   x

3 2 + c

11) If g1(x) = 4x3 + 3x2 + 6x and g(1) = -3, then g(x) =

A) x4 + x3 + 6x2 - 11

B) + + 3x2 -

C) 12x2 + 6x + 6

D) x4 + x3 + 3x2

E) x4 + x3 + 3x2 - 8

12) Which of the following defines a function f such

that f1(x) =   x and the graph of function f passthrough the point (9,0)?

A) f(x) = x  x - 18

B) f(x) = x  x - 3x

C) f(x) = + 9

D) f(x) =   x - 3

E) f(x) = x  x - 18

13) The slope of a curve at each point (x,y) is given by4x - 1. Which of the following is an equation for this

curve if it passes through the point (-2,3)?

A) y = 2x2 - x - 7

B) y = 4x2 - x - 15

C) y = 2x2 - x + 7

D) y = x2 - 4x - 9

E) y = 2x2 - x

14) If function f has a derivative defined by

f1(x) = and f(1) = 0, then f(4) =

A)

B) *

C)

D) *

E)

15) The slope of the line tangent to the graph of a

function f at any point (x,y) is given by x3 - x. Ifthe graph of function f passes through the point

(2,1), find f(0).

A) 1

B) 2

C) 3

D) -1

E) 0

16) A function f has a derivative f1(x) = 3 - 2x. Anequation of the line tangent to the graph of

function f at x = 2 is y - 7 = -(x - 2). What is an

equation of function f?

A) f(x) = -x2 + 3x

B) f(x) = -3x2 + x - 3

C) f(x) = -x2 + 3x - 3

D) f(x) = 3x2 + 3x - 1

E) f(x) = x2 - 3x + 3

17) If h2(x) = x - 2, h1(4) = 0, and h(0) = 4, then h(x) =

A) - x2 + 4

B) 2x3 - x2 + 4

C) - + 4

D) - x2

E) x2 + 2x + 4

18) At each point (x,y) on a curve, = 6x.

Additionally, the line y = 6x + 4 is tangent to thecurve at x = -2. Which of the following is an

equation of the curve that satisfies these

conditions?

A) y = 6x2 - 32

B) y = x3 - 6x - 12

C) y = 2x3 - 3x

D) y = x3 - 6x + 12

E) y = 2x3 + 3x - 12

ANTIDERIVATIVES OF TRIG FUNCTIONS & INITIALV VALUE PROBLEMS PRACTICE

1)

A) * + 2 sin t + c

B) * - 2 sin t + c

C) * - 2 sin t + c

D) * + 2 sin t + c

E) * + 2 sin t + c

2)

A) -cot x + c

B) -sin x + c

C) cos x + c

D) + c

E) -cos x + c

3)

A) -2 cos t + c

B) cos t + c

C) cos 2t + c

D) 2 cos t + c

E) -cos 2t + c

4)

A) sec x + cos x + c

B) -csc x + c

C) -sec x + tan x + c

D) -csc x + x + c

E) csc x + x + c

5)

A) tan x - x + c

B) tan x + x + c

C) tan x + c

D) sec x - x + c

E) sec x + x + c

6) If f1(x) = 2 cos x - 3 sin x and f(0) = 4, then f(x) =

A) 2 sin x + 3 cos x + 1

B) 2 sin x + 3 cos x

C) 2 cos x - 3 sin x + 2

D) 3 sin x - 2 cos x + 6

E) 3 cos x - 2 sin x + 1

7) The derivative of a function f is given by

f1(x) = sec2 x + cos x. If f( ) = 1, then f(x) =A) tan x + sin x +

B) tan x - sin x +

C) tan x + sin x +

D) tan x + sin x -

E) tan x - sin x -

8) Function f has a second derivative that is given by

f2(x) = x + cos x. Which of the following could be

f(x)?

A) - 3x + cos x - 1

B) + cos x - 1

C) - 3x - cos x - 1

D) - 3x - sin x - 1

E) - 3x + sin x + 1

9) Function g has a second derivative that is given by

g2(x) = x2 - sin x. Which of the following could be

function g?

A) + cos x - x + 1

B) + sin x - x

C) - cos x + x - 1

D) - sin x - x + 1

E) - sin x - x + 1

10) The slope of a curve at each point (x,y) is given by2 cos x - x. Which of the following is an equation of

this curve if its graph passes through the point (0,1)?

A) 2 sin x - x2 + 1

B) cos2 x - x2 + 1

C) cos2 x - + 1

D) 2 sin x - + 1

E) -2 sin x - + 1

Questions 11 through 17 refer to the following:

Evaluate the given integral.

11)

12)

13)

14)

15)

16)

17)

©i u2w05113h LKquKt0ag dSSozfEtwwDalrUes CLsLbCc.X s gAKlPlS QrJiqg6hEtusw 8rceSsuerrPvmeJdZ.n T SM9a2dReN RwmiutSh8 QIdnFfpi2ntiRtkeU FCeaAlucGuIlqu7sk.I Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus Name___________________________________

Period____Date________________Integration - Logarithmic Rule and Exponentials

Evaluate each indefinite integral.

1) ∫ x−1

dx 2) ∫ 3x−1

dx

3) ∫ −1

xdx 4) ∫

1

xdx

5) ∫ −ex

dx 6) ∫ ex

dx

7) ∫ 2 ⋅ 3x

dx 8) ∫ 3 ⋅ 5x

dx

©s K2P0h1u3X wK4uvtOam xSgo6fMtcwLa8r7ew ELpLCCt.s L SAgl1lA HrOi4gihjtJsv RrTecsCeurkvKe2d5.Q G jMLa1d8ep JwLiGtvhF 9IHnjfBiMnEiQtPei XCYaklEcsuXlCupsM.Y Worksheet by Kuta Software LLC

Integration Inverse functionEvaluate each indefinite integral.

1) ∫1

16 − x2

dx 2) ∫1

4 + x2

dx

3) ∫1

x x2 − 1

dx 4) ∫1

16 + x2

dx

5) ∫1

x x2 − 4

dx 6) ∫1

25 − x2

dx

7) ∫1

x x2 − 81

dx 8) ∫1

4 + x2

dx